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As my question states, I am wondering if there is any chance we use the prior predictive distribution. I am studying Bayesian Statistics and have understood what it is. It is a must to go through in Bayesian Statistics and seems very useful(at least to my understanding). My understanding about it is that it is used before seeing data to see what the distribution of observations might be. But I still cannot understand if we ever use it. Why would we use it? Is this for a simulation purpose? I wonder if you have ever used or if you are using it.

mathccino
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  • Can you provide a reference? Never heard about prior predictive distributions… – utobi Jan 18 '24 at 05:23
  • Well, we use it all the time, the prior predictive distribution being another word (or at least very very closely related to) for the marginal likelihood, which is for example used when comparing models via Bayes factors. https://stats.stackexchange.com/questions/394648/differences-between-prior-distribution-and-prior-predictive-distribution https://stats.stackexchange.com/questions/147513/marginal-likelihood-vs-prior-predictive-probability https://stats.stackexchange.com/questions/130832/explanation-that-the-prior-predictive-marginal-distribution-follows-from-prior – Christoph Hanck Jan 18 '24 at 07:50
  • @ChristophHanck tanks for the reply. I’ve done some work on marginal likelihood but the term prior predictive distribution is new to me. – utobi Jan 18 '24 at 20:03
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    Personally, I prefer to select the parameters of the prior distribution by looking at the prior predictive distribution. Because it is more natural to have a guess regarding the data rather than a guess regarding the parameters. – Stéphane Laurent Jan 24 '24 at 18:02

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The simplest answer is that it does not matter in the vast majority of cases. You can just assume you have a completely flat prior. Then let the data speak for itself.

In very rare circumstances it might matter. Usually that happens when you have very limited data. But, if so, then any statistical analysis used is suspicious anyway.

Why do you need the prior? Because by Bayes Theorem, P( hypothesis | data ) is proportional to P( data | hypothesis )P( hypothesis ). As you can see, you need the quantity P( hypothesis ) in the formula. Therefore, it is a necessary part of the formula.

Nicolas Bourbaki
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  • I know the prior is needed but my question is that I want to know if we ever use the prior predictive distribution. If you think I haven't understood your answer, please stretch it a bit further. – mathccino Jan 18 '24 at 04:39
  • @mathlover The quick answer is "no". But, some people might tell you, that after you pick your prior distribution, it might be a good idea to run a random simulation using it to see if the data generated makes sense. For example, let us say your prior distribution results in people having heights of negative-inches, then that looks suspicious. So it might be a good idea to just run a quick simulation to see how sensible your prior is before you use it. – Nicolas Bourbaki Jan 18 '24 at 04:51
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    @NicolasBourbaki, your answer reads (at least to me) as though you discuss the prior and not the prior predictive distribution. As the links above discuss, they however are two different things. – Christoph Hanck Jan 18 '24 at 09:29